arXiv subjects

Dimiter Prodanov

Publications and source records attributed to Dimiter Prodanov.

21 records · Page 2Linked to original sources

arXiv2015

Regularized and Fractional Taylor expansions of Holderian functions

Holderian functions have strong non-linearities, which result in singularities in the derivatives. This manuscript presents several fractional-order Taylor expansions of Hölderian functions around points of non- differentiability. These expansions are derived using the concept of a fractional velocity, which can be used to describe the singular behavior of derivatives and non-differentiable functions. Fractional velocity is defined as the limit of the difference quotient of the increment of a function and the difference of its argument raised to a fractional power. Fractional velocity can be used to regularize ordinary derivatives. To this end, it is possible to define regularized Taylor series and compound differential rules. In particular a compound differential rule for Holder 1/2 functions is demonstrated. The expansion is presented using the auxiliary concept of fractional co-variation of functions.

math.CA↗

arXiv2015

Fractional variation of Hölderian functions

The paper demonstrates the basic properties of the local fractional variation operators (termed fractal variation operators). The action of the operators is demonstrated for local characterization of Holderian functions. In particular, it is established that a class of such functions exhibits singular behavior under the action of fractal variation operators in infinitesimal limit. The link between the limit of the fractal variation of a function and its derivative is demonstrated. The paper presents a number of examples, including the calculation of the fractional variation of Cauchy sequences leading to the Dirac's delta-function.

math.CA↗

arXiv2015

Product rules for the Fractional variation of Hölderian functions

Fractional variation is defined as the limit of the difference quotient of the increments of a function and its argument raised to a fractional power. Fractional velocity can be suitable for characterizing singular behavior of derivatives of Hölderian functions and non differentiable functions. The manuscript derives the product rules for fractional variation. Correspondence with integer-order derivatives is discussed. It is demonstrated that for Hölder functions under certain conditions the product rules deviates from the Leibniz rule. This deviation is expressed by another quantity, fractional co-variation. Basic algebraic properties of the fractional co-variation are demonstrated.

math.CA↗